Associate Professor, Mathematics

I'm a
low-dimensional topologist specializing in the topology of smooth and symplectic 4-manifolds, using Lefschetz fibrations and tools from Heegaard Floer homology and gauge theory. I arrived here at Virginia in the fall of 2006.

My current work is focused on the study of smooth
3- and 4-dimensional manifolds, and contact and symplectic structures on them, using Lefschetz fibrations and related structures. Central tools are Heegaard Floer homology and the associated invariants of 4-manifolds, as well as techniques from symplectic topology and geometry.

The following is a list of my preprints and publications:

"Convex plumbing and Lefschetz fibrations," with David Gay. To appear in Journal of Symplectic Geometry. We show that many operations arising from monodromy substitution can be performed naturally in the symplectic category.

"Knotted surfaces in 4-manifolds." To appear in Forum Mathematicum. We extend a result of Fintushel and Stern to show that given a symplectic surface with simply-connected complement and self-intersection at least 2-2g in a symplectic 4-manifold with b2+ >1, there are infinitely many embedded surfaces topologically isotopic to the original but not smoothly isotopic to it.

"Monodromy substitution and rational blowdowns," with H. Endo and J. Van Horn-Morris. Journal of Topology4 (2011) 227--253. We exhibit several new families of relations in mapping class groups of planar surfaces that give rise to various rational blowdowns under the operation of monodromy substitution. (Arxiv version)

"Product formulae for Ozsvath-Szabo 4-manifold invariants," with S.
Jabuka. Geometry and Topology12 (2008) 1557--1651. We develop a general formalism for calculating Ozsvath-Szabo invariants for closed 4-manifolds obtained by gluing two manifolds with boundary, in terms of relative invariants of the pieces. The approach makes use of "perturbed" Heegaard Floer homology, that is, Floer homology with coefficients in modules over certain Novikov rings. As a motivating example and illustration of the formalism, we determine the behavior of Ozsvath-Szabo 4-manifold invariants under fiber sums of manifolds along surfaces with trivial normal bundle. (Arxiv version)

"On the Heegaard Floer
homology of a surface times a circle," with S. Jabuka. Advances in Mathematics218 (2008) 728--761. We compute the Heegaard Floer homology of the 3-manifold named in the title, and show in particular that the integral Floer homology of the product of a surface of genus at least 3 with the circle contains nontrivial torsion. (Arxiv version)